System for analyzing sensitivity of parasitic capacitance to variation of semiconductor design parameters

ABSTRACT

A sensitivity analysis system has a memory device in which an interconnect structure data indicating an interconnect structure included in a semiconductor device is stored. The interconnect structure has: a main interconnection; and a contact structure electrically connected to the main interconnection and extending toward a semiconductor substrate. Parameters contribute to parasitic capacitance of the interconnect structure, and variation of each parameter from a design value caused by manufacturing variability is represented within a predetermined range. The sensitivity analysis system further has: a parameter setting unit that sets the variation to a plurality of conditions within the predetermined range; a capacitance calculation unit that calculates the parasitic capacitance of the interconnect structure in each of the plurality of conditions; and a sensitivity analysis unit that analyzes, based on the calculated parasitic capacitance, response of the parasitic capacitance to variation of the each parameter.

INCORPORATION BY REFERENCE

This application is based upon and claims the benefit of priority fromJapanese patent application No. 2009-006504, filed on Jan. 15, 2009, thedisclosure of which is incorporated herein in its entirety by reference.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a design technique for a semiconductordevice. In particular, the present invention relates to a designtechnique for a semiconductor device having a contact structure.

2. Description of Related Art

In a manufacturing process of a semiconductor device, an interconnectstructure may not be manufactured as expected. That is, physicalparameters such as a width and a thickness of an interconnection, athickness of an interlayer insulating film and the like may vary fromtheir desired design values. Such manufacturing variability affectsdelay in a circuit. Thus, even if a designed circuit passes timingverification on a computer, an actual product may malfunction since themanufacturing variability occurs. Therefore, it is desirable to performthe timing verification in consideration of the manufacturingvariability (refer, for example, to Japanese Laid-Open PatentApplication JP-2007-172258).

Meanwhile, to consider the manufacturing variability during the timingverification means that a condition to be met in the timing verificationbecomes stricter. As the condition becomes stricter, the timingverification is more likely to result in fail and thus the number ofcircuit design modification times increases. This causes increase in adesign TAT (Turn Around Time).

Japanese Laid-Open Patent Application JP-2006-209702 discloses atechnique that can suppress increase in the design TAT while consideringthe manufacturing variability. According to the technique, unrealisticpatterns of the manufacturing variability are excluded fromconsideration. For example, let us consider a case where a width and athickness of an interconnection can vary from respective design valuesin a range from −3σ to +3σ (σ: standard deviation). In this case, aprobability that both the width and thickness “simultaneously” vary tothe maximum extent is extremely low from a statistical point of view. Ifsuch extreme situations are taken into consideration, it is necessary tosupport those extreme situations, which causes increase in the number ofcircuit design modification times. Therefore, according to thetechnique, such the extreme situations are excluded from theconsideration (this scheme is hereinafter referred to as “statisticalrelaxation”. More specifically, the statistical relaxation is applied tocalculation of corner conditions under which an interconnect delaybecomes maximum or minimum. Then, interconnect resistance andinterconnect capacitance under the corner conditions are provided as alibrary. This library is referred to in LPE (Layout ParameterExtraction). Consequently, it is possible to perform the timingverification in consideration of the manufacturing variability whileexcluding the extreme situations. In other words, it is possible toperform high-accuracy timing verification while preventing unnecessaryincrease in the design TAT.

Japanese Laid-Open Patent Application JP-2008-028161 also discloses amethod of designing a semiconductor device in consideration of thestatistical relaxation. First, correction parameters indicatingvariations of interconnect resistance and parasitic capacitance fromdesign values due to the manufacturing variability are calculated. Atthis time, the correction parameters are calculated based on theabove-described statistical relaxation scheme. Next, the LPE isperformed based on a layout of the semiconductor device and thereby theinterconnect resistance and parasitic capacitance related to aninterconnection in the layout are extracted. After the LPE is completed,the extracted interconnect resistance and parasitic capacitance arerespectively corrected by using the above-mentioned correctionparameters. The post-correction interconnect resistance and parasiticcapacitance are used for operation verification of the semiconductordevice.

By the way, a contact structure is used in a semiconductor device forconnecting between an interconnection formed in an interconnection layerand a transistor formed on a semiconductor substrate. The contactstructure is so formed as to penetrate through an interlayer insulatingfilm between the interconnection and the transistor.

Here, the inventor of the present application has recognized thefollowing points. With increasing miniaturization of a semiconductordevice in recent years, an interval between adjacent contact structuresis getting narrower. Such tendency is conspicuous, for example, in amemory macro in which a large number of cell transistors are integrated.When the interval between adjacent contact structures becomes narrower,influence of the manufacturing variability of the contact structure onthe parasitic capacitance is considered to become significant andnonnegligible. It is therefore desirable in designing a semiconductordevice to consider the influence of the manufacturing variability of theinterconnection as well as the contact structure.

SUMMARY OF THE INVENTION

In one embodiment of the present invention, a sensitivity analysissystem is provided. The sensitivity analysis system has a memory device,a parameter setting unit, a capacitance calculation unit and asensitivity analysis unit. An interconnect structure data indicating aninterconnect structure included in a semiconductor device is stored inthe memory device. Here, the interconnect structure has: a maininterconnection formed in an interconnection layer; and a contactstructure electrically connected to the main interconnection andextending from the main interconnection toward a semiconductorsubstrate. A plurality of parameters contribute to parasitic capacitanceof the interconnect structure. Variation of each of the plurality ofparameters from a design value caused by manufacturing variability isrepresented within a predetermined range.

The parameter setting unit sets the variation of each parameter to aplurality of conditions within the predetermined range. The capacitancecalculation unit calculates the parasitic capacitance of theinterconnect structure in each of the plurality of conditions. Based onthe calculated parasitic capacitance, the sensitivity analysis unitanalyzes response of the parasitic capacitance to variation of the eachparameter.

According to the present invention, it is possible to analyze influenceof the manufacturing variability of the contact structure on theparasitic capacitance.

BRIEF DESCRIPTION OF THE DRAWINGS

The above and other objects, advantages and features of the presentinvention will be more apparent from the following description ofcertain preferred embodiments taken in conjunction with the accompanyingdrawings, in which:

FIG. 1 is a schematic diagram showing an interconnect structureincluding a contact structure;

FIG. 2 is a circuit diagram showing a SRAM cell array;

FIG. 3 is a plan view showing an example of a planar layout of onememory cell;

FIG. 4 is a schematic diagram showing the interconnect structure in theSRAM;

FIG. 5 shows an example of setting of variation of each parameter in acase of sensitivity analysis with respect to a word line;

FIG. 6 is a graph showing a result of the sensitivity analysis withrespect to the word line;

FIG. 7 shows an example of setting of variation of each parameter in acase of sensitivity analysis with respect to a bit line;

FIG. 8 is a graph showing a result of the sensitivity analysis withrespect to the bit line;

FIG. 9 is a block diagram showing a configuration of a sensitivityanalysis system according to an embodiment of the present invention;

FIG. 10 is a flow chart showing a method of the sensitivity analysisaccording to the embodiment of the present invention;

FIG. 11 is a conceptual diagram for explaining the statisticalrelaxation;

FIG. 12 is a conceptual diagram showing an interconnect structure andparameters contributing to the parasitic capacitance of the interconnectstructure;

FIG. 13 shows an example of setting of variation of each parameter in acase of capacitance analysis with respect to the interconnect structure;

FIG. 14 is a graph showing variation in the parasitic capacitance of theinterconnect structure;

FIG. 15 is a conceptual diagram showing an interconnect structure andparameters contributing to the interconnect resistance of theinterconnect structure;

FIG. 16 is a conceptual diagram showing a method of calculating aresistance value of the contact structure;

FIG. 17 shows an example of setting of variation of each parameter in acase of resistance analysis with respect to the interconnect structure;

FIG. 18 is a graph showing variation in the interconnect resistance ofthe interconnect structure;

FIG. 19 shows an example of CR combination in a CR-added cell netlist;

FIG. 20 is a block diagram showing a configuration of a CR extractionsystem according to the embodiment of the present invention;

FIG. 21 is a flow chart showing a method of CR extraction according tothe embodiment of the present invention;

FIG. 22 is a schematic diagram showing a configuration of a SRAM macro;

FIG. 23 is a block diagram showing a configuration of a circuit designsystem according to the embodiment of the present invention; and

FIG. 24 is a flow chart showing a method of designing and a method ofmanufacturing the semiconductor device according to the embodiment ofthe present invention.

DESCRIPTION OF PREFERRED EMBODIMENTS

The invention will be now described herein with reference toillustrative embodiments. Those skilled in the art will recognize thatmany alternative embodiments can be accomplished using the teachings ofthe present invention and that the invention is not limited to theembodiments illustrated for explanatory purposed.

1. Equipotential Interconnect Structure Including Contact Structure

FIG. 1 schematically shows an interconnect structure 1 in asemiconductor device, which is considered in the present embodiment. InFIG. 1, a plane parallel to a surface of a semiconductor substrate isrepresented by a XY plane and a direction perpendicular to the XY planeis represented by a Z-direction.

As shown in FIG. 1, the interconnect structure 1 includes a maininterconnection 10 and a contact structure 20 electrically connected tothe main interconnection 10. The main interconnection 10 is formed in aninterconnection layer located above the semiconductor substrate and isparallel to the XY plane. On the other hand, the contact structure 20 isso formed along the Z-direction as to penetrate through an interlayerinsulating film 5 below the main interconnection 10. That is, thecontact structure 20 is so formed as to extend from the maininterconnection 10 toward the semiconductor substrate. The contactstructure 20 has a “stacked structure” in which conductors (vias,cushion interconnections, contacts and the like) formed in respectivelayers below the main interconnection 10 are stacked. The cushioninterconnection is an interconnection in the stacked structure.

The main interconnection 10 and the contact structure 20 areelectrically connected with each other, and their potentials are thesame. In that sense, it can be said that the interconnect structure 1 isan “equipotential structure”. Regarding a netlist used in a designphase, the equipotential interconnect structure 1 is represented by onenet (equipotential net).

Devices having the interconnect structure 1 as shown in FIG. 1 includesemiconductor memories such as DRAM, SRAM, eDRAM (embedded DRAM) andflash memory. Such a semiconductor memory is provided with word linesand bit lines for selectively drive a cell transistor in a memory cell.A word line and a bit line respectively are connected to the celltransistor through contact structures. Therefore, the word line or thebit line corresponds to the main interconnection 10 shown in FIG. 1.

FIG. 2 shows a circuit configuration of an SRAM as an example of thesemiconductor memory. The SRAM has a cell array in which a plurality ofmemory cells CELL are arranged in an array form. Moreover, a pluralityof word lines WD are formed in the X-direction, and a plurality ofcomplementary bit line pairs BIT, /BIT are formed in the Y-direction.The memory cells CELL are placed at respective intersections of the wordlines WD and the complementary bit line pairs BIT, /BIT. Each memorycell CELL has a memory unit comprising cell transistors TR1, TR2 and twoinverters. Gates of the cell transistors TR1 and TR2 are connected toone word line WD. A source or a drain of the cell transistor TR1 isconnected to the bit line BIT, and a source or a drain of the celltransistor TR2 is connected to the bit line /BIT. The cell transistorsTR1 and TR2 are driven by applying appropriate potentials respectivelyto the word line WD and the bit lines BIT and /BIT.

FIG. 3 shows an example of a planar layout of one memory cell CELL shownin FIG. 2. The word line WD is connected to the cell transistor TR1(TR2) through the contact structure 20-1 (20-2).

FIG. 4 shows a three-dimensional structure of one unit including oneword line WD and one bit line BIT shown in FIG. 3. As shown in FIG. 4,interconnection layers M1, M2, M3 and M4 are provided in this order frombelow to above. Interconnections extending in the X-direction are formedin the interconnection layers M1 and M3. On the other hand,interconnections extending in the Y-direction are formed in theinterconnection layers M2 and M4.

The word line WD is the main interconnection 10 formed in theinterconnection layer M3, which extends in the X-direction. The wordline WD is connected to the cell transistor TR through the contactstructure 20. The word line WD and the contact structure 20 constituteone interconnect structure 1 (equipotential structure). Here, thecontact structure (stacked structure) 20 connected to the word line WDincludes the gate of the cell transistor TR, a contact formed betweenthe gate and the interconnection layer M1, a cushion interconnectionformed in the interconnection layer M1, a via formed between theinterconnection layers M1 and M2, a cushion interconnection formed inthe interconnection layer M2 and a via formed between theinterconnection layers M2 and M3.

The bit line BIT is the main interconnection 10 formed in theinterconnection layer M2, which extends in the Y-direction. The bit lineBIT is connected to the cell transistor TR through the contact structure20. The bit line BIT and the contact structure 20 constitute oneinterconnect structure 1 (equipotential structure). Here, the contactstructure (stacked structure) 20 connected to the bit line. BIT includesa contact formed between the source/drain and the interconnection layerM1, a cushion interconnection formed in the interconnection layer M1 anda via formed between the interconnection layers M1 and M2.

In the cell array of the SRAM, the unit structure shown in FIG. 4 arerepeated. By using the unit structure shown in FIG. 4 as an example,extraction of parasitic capacitance and interconnect resistance of theinterconnect structure 1 will be described below in detail.

2. Sensitivity Analysis

According to the present embodiment, as will be described later,extraction of the parasitic capacitance C of the interconnect structure1 is performed in consideration of the manufacturing variability. Theparasitic capacitance C of the interconnect structure 1 depends on aplurality of parameters Pk (index k represents kind of the parameters).In other words, the plurality of parameters Pk contribute to theparasitic capacitance C of the interconnect structure 1.

The parameter Pk can be expressed as “Pk=P0 k+ΔPk”. Here, P0 k is adesign value (typical value) of the parameter Pk, and ΔPk is variationfrom the design value P0 k caused by the manufacturing variability. Thevariation ΔPk is represented within a predetermined variation range“−δPk to +δPk”. Here, −δPk is a lower limit of the variation ΔPk, and+δPk is an upper limit of the variation ΔPk. Typically, a distributionof the variation ΔPk is represented by a normal distribution. When thestandard deviation of the distribution is σPk, the variation ΔPk can beexpressed as “ΔPk=σPk×σPk”. When the coefficient σPk is within a rangefrom −3 to +3, the range of the variation ΔPk is represented by “−3σPkto +3σPk”, which is sufficient from a statistical point of view.

In order to extract the parasitic capacitance C of the interconnectstructure 1 in consideration of the manufacturing variability, it isnecessary to vary each parameter Pk within the above-mentioned variationrange. However, to blindly vary all the parameter Pk is inefficient,which causes significant increase in computation load. It is thereforeimportant to previously determine policy on how to and which parameterPk has to be treated. To this end, according to the present embodiment,sensitivity (response) of the parasitic capacitance C to manufacturingvariation of each parameter Pk is first analyzed.

For example, whether the parasitic capacitance C increases or decreasesin response to variation of a certain parameter Pk from its design valueis analyzed.

Also for example, to what extent variation of a certain parameter Pkcontributes to variation of the parasitic capacitance C, namely, degreeof contribution of the parameter Pk to the parasitic capacitance C isanalyzed. A parameter Pk whose degree of contribution to the parasiticcapacitance C is relatively large is hereinafter referred to as a “mainparameter (critical parameter)”. On the other hand, a parameter Pk whosedegree of contribution to the parasitic capacitance C is relativelysmall is hereinafter referred to as a “sub parameter”. To classify theparameters Pk into the main parameter and the sub parameter is importantin terms of efficient extraction processing of the parasitic capacitanceC. For example, the manufacturing variability of the main parameter isconsidered in depth while the manufacturing variability of the subparameter is considered simply, which can improve the processingefficiency (computing speed).

It should be noted that since the main interconnection 10 (word line WD,bit line BIT) occupies the largest volume among components of theinterconnect structure 1, at least a width and thickness of the maininterconnection 10 are treated as the “main parameter”. In thesensitivity analysis, whether another parameter Pk is the main parameteror the sub parameter is determined. In particular, the interconnectstructure 1 in the present embodiment includes the contact structure 20and the parameters Pk related to the contact structure 20 are analyzedintensively.

The sensitivity analysis of the parasitic capacitance C of a wordinterconnect structure having the word line WD as the maininterconnection 10 and the sensitivity analysis of the parasiticcapacitance C of a bit interconnect structure having the bit line BIT asthe main interconnection 10 will be described below.

2-1. Word Interconnect Structure

The sensitivity analysis of the parasitic capacitance C of the wordinterconnect structure will be described with reference to FIG. 4 andFIG. 5. The word line WD as the main interconnection 10 is formed in theinterconnection layer M3. A width and a thickness of the word line WDare w3 and t3, respectively. A film thickness of an interlayerinsulating film formed in a layer V3 (first layer) placed immediatelyabove the interconnection layer M3 is d3. A layer placed immediatelyabove the layer V3 is the interconnection layer M4(supra-interconnection layer), and a width and a thickness of aninterconnection formed in the interconnection layer M4 are w4 and t4,respectively. The thickness t4 is the same as a film thickness of aninterlayer insulating film in the interconnection layer M4.

The contact structure 20 has the stacked structure in which conductorsformed in respective layers below the interconnection layer M3 arestacked. A film thickness of an interlayer insulating film formed in alayer V2 (second layer) placed immediately below the interconnectionlayer M3 is d2. A layer placed immediately below the layer V2 is theinterconnection layer M2 (sub-interconnection layer), and a width and athickness of an interconnection formed in the interconnection layer M2are w2 and t2, respectively. The thickness t2 is the same as a filmthickness of an interlayer insulating film in the interconnection layerM2. A film thickness of an interlayer insulating film formed in a layerV1 placed immediately below the interconnection layer M2 is d1. A layerplaced immediately below the layer V1 is the interconnection layer M1,and a width and a thickness of an interconnection formed in theinterconnection layer M1 are w1 and t1, respectively. The thickness t1is the same as a film thickness of an interlayer insulating film in theinterconnection layer M1. A film thickness of an interlayer insulatingfilm below the interconnection layer M1 is d0.

The above-mentioned w4, t4, d3, w3, t3, d2, w2, t2, d1, t1 and d0 areconsidered as a plurality of parameters Pk contributing to the parasiticcapacitance C of the word interconnect structure. The interconnect widthw3 and the interconnect thickness t3 of the word line WD among them arethe main parameters. The variation ΔPk of each parameter Pk from itsdesign value is represented within a variation range “−3σPk to +3σPk”.

In the present embodiment, TCAD (Technology CAD) is used to perform“three-dimensional electromagnetic analysis” of the unit structure shownin FIG. 4 in order to calculate the parasitic capacitance C of the wordinterconnect structure. By the three-dimensional electromagneticanalysis with the use of the TCAD, the parasitic capacitance C can becalculated with high accuracy. In the analysis, the variation ΔPk ofeach parameter Pk contributing to the parasitic capacitance C is set tovarious conditions within the above-mentioned variation range.

FIG. 5 shows seven kinds of examples as the set conditions of thevariation ΔPk of each parameter Pk. In each condition, “+3σ” means thatthe variation ΔPk of the corresponding parameter Pk is set to the upperlimit (+3σPk), and “−3σ” means that the variation ΔPk of thecorresponding parameter, Pk is set to the lower limit (−3σPk). Nothingdescribed means that the variation ΔPk is 0, namely the correspondingparameter Pk is set to its design value P0 k.

Condition-1: All parameters Pk are set to the respective design valuesP0 k (center condition). The parasitic capacitance C calculated at thistime is Ctyp.

Condition-2: The variation of the film thickness d3 immediately abovethe interconnection layer M3 and the film thickness d2 immediately belowthe interconnection layer M3 each is set to the upper limit (+3σ).

Condition-3: The variation of the film thickness d3 immediately abovethe interconnection layer M3 and the film thickness d2 immediately belowthe interconnection layer M3 each is set to the lower limit (−3σ).

Condition-4: The variation of the film thickness d3 immediately abovethe interconnection layer M3 and the film thicknesses (d2, t2, d1, t1and d0) of respective layers of the contact structure 20 each is set tothe upper limit (+3σ). That is, the variation of t2, d1, t1 and d0 eachis set to the upper limit, in addition to the above-mentionedCondition-2.

Condition-5: The variation of the film thickness d3 immediately abovethe interconnection layer M3 and the film thicknesses (d2, t2, d1, t1and d0) of respective layers of the contact structure 20 each is set tothe lower limit (−3σ). That is, the variation of t2, d1, t1 and d0 eachis set to the lower limit, in addition to the above-mentionedCondition-3.

Condition-6: The variation of the film thickness d3 immediately abovethe interconnection layer M3 and the film thickness d2 immediately belowthe interconnection layer M3 each is set to the lower limit (−3σ), as inthe case of the above Condition-3. In addition to that, the variation ofthe interconnect width w4 in the supra-interconnection layer M4 and theinterconnect width w2 in the sub-interconnection layer M2 each is set tothe upper limit (+3σ).

Condition-7: The variation of the film thickness d3 immediately abovethe interconnection layer M3 and the film thickness d2 immediately belowthe interconnection layer M3 each is set to the lower limit (−3σ), as inthe case of the above Condition-3. In addition to that, the variation ofthe interconnect thickness t4 in the supra-interconnection layer M4 andthe interconnect thickness t2 in the sub-interconnection layer M2 eachis set to the upper limit (+3σ).

FIG. 6 shows calculation results of the parasitic capacitance C withrespect to the above-mentioned conditions. The vertical axis representsa variation ratio (C−Ctyp)/Ctyp of the parasitic capacitance C. Thefollowings can be seen from the results shown in FIG. 6.

It turns out from a comparison between Condition-1 and Condition-2 thatthe parasitic capacitance C decreases when the film thickness d3immediately above the interconnection layer M3 and the film thickness d2immediately below the interconnection layer M3 are increased. Also, itturns out from a comparison between Condition-1 and Condition-3 that theparasitic capacitance C increases when the film thickness d3 immediatelyabove the interconnection layer M3 and the film thickness d2 immediatelybelow the interconnection layer M3 are decreased. That is to say, itturns out that the parasitic capacitance C is in inverse proportion tothe film thickness d2, d3.

It turns out from a comparison between Condition-2 and Condition-4 thatthe parasitic capacitance C increases when the film thicknesses t2, d1,t1 and d0 in the contact structure 20 are increased. Also, it turns outfrom a comparison between Condition-3 and Condition-5 that the parasiticcapacitance C decreases when the film thicknesses t2, d1, t1 and d0 inthe contact structure 20 are decreased. That is to say, it turns outthat the parasitic capacitance C is proportional to the film thicknessest2, d1, t1 and d0. Note that this trend is opposite to theabove-mentioned trend regarding the film thickness d2, d3. This trend isattributed to a fact that lateral capacitance of the contact structure20 is increased when the film thicknesses t2, d1, t1 and d0 areincreased.

In the cases of Condition-4 and Condition-5, the variation of theparasitic capacitance C from Ctyp is small and the absolute value of thevariation ratio is at most 1%. This is because, even when the filmthickness of each layer is increased (decreased) as a whole, decrease(increase) in the parasitic capacitance C due to the increase (decrease)in d3 and d2 and increase (decrease) in the parasitic capacitance C dueto the increase (decrease) in t2, d1, t1 and d0 cancel each other.

It turns out from a comparison between Condition-3 and Condition-6 thatthe variation of the interconnect width w4 in the supra-interconnectionlayer M4 and the interconnect width w2 in the sub-interconnection layerM2 has little effect on the parasitic capacitance C. Based on thisresult, it is possible to determine that the manufacturing variabilityof the interconnect widths w4 and w2 need not be considered onextracting the parasitic capacitance C.

It turns out from a comparison between Condition-3 and Condition-7 thatthe parasitic capacitance C increases when the interconnect thickness t4in the supra-interconnection layer M4 and the interconnect thickness t2in the sub-interconnection layer M2 are increased. Note that this trendalso is opposite to the above-mentioned trend regarding the filmthickness d2, d3. The increase in the parasitic capacitance C in thiscase is mainly attributed to increase in lateral capacitance of thecontact structure 20 due to the increase in the interconnect thicknesst2 in the sub-interconnection layer M2. On the other hand, influence ofthe increase in the interconnect thickness t4 in thesupra-interconnection layer M4 is considered to be small.

Moreover, it turns out that the variation of the parasitic capacitance Cfrom Ctyp is not so much in either condition. Let us consider a casewhere a threshold value of nonnegligible variation in the parasiticcapacitance C is “absolute value of the variation ratio=3%”, forexample. In this case, it turns out that the absolute value of thevariation ratio is less than the threshold value in either condition.Based on this result, it is possible to determine that the parametersother than the interconnect width w3 and the interconnect thickness t3of the word line WD can be treated as the “sub parameters”.

2-2. Bit Interconnect Structure

The sensitivity analysis of the parasitic capacitance C of the bitinterconnect structure will be described with reference to FIG. 4 andFIG. 7. The bit line BIT as the main interconnection 10 is formed in theinterconnection layer M2. A width and a thickness of the bit line BITare w2 and t2, respectively. A film thickness of an interlayerinsulating film formed in a layer V2 (first layer) placed immediatelyabove the interconnection layer M2 is d2. A layer placed immediatelyabove the layer V2 is the interconnection layer M3(supra-interconnection layer), and a width and a thickness of aninterconnection formed in the interconnection layer M3 are w3 and t3,respectively. The thickness t3 is the same as a film thickness of aninterlayer insulating film in the interconnection layer M3. Filmthicknesses of interlayer insulating films in the layer V3 above theinterconnection layer M3 and the interconnection layer M4 are d3 and t4,respectively.

The contact structure 20 has the stacked structure in which conductorsformed in respective layers below the interconnection layer M2 arestacked. A film thickness of an interlayer insulating film formed in alayer V1 (second layer) placed immediately below the interconnectionlayer M2 is d1. A layer placed immediately below the layer V1 is theinterconnection layer M1 (sub-interconnection layer), and a width and athickness of an interconnection formed in the interconnection layer M1are w1 and t1, respectively. The thickness t1 is the same as a filmthickness of an interlayer insulating film in the interconnection layerM1. A film thickness of an interlayer insulating film below theinterconnection layer M1 is d0.

The above-mentioned t4, d3, w3, t3, d2, w2, t2, d1, w1, t1 and d0 areconsidered as a plurality of parameters Pk contributing to the parasiticcapacitance C of the bit interconnect structure. The interconnect widthw2 and the interconnect thickness t2 of the bit line BIT among them arethe main parameters. The variation ΔPk of each parameter Pk from itsdesign value is represented within a variation range “−3σPk to +3σPk”.

As in the case of the word interconnect structure, the TOAD is used toperform “three-dimensional electromagnetic analysis” of the unitstructure shown in FIG. 4. Thus, the parasitic capacitance C of the bitinterconnect structure can be calculated with high accuracy. In theanalysis, the variation ΔPk of each parameter Pk contributing to theparasitic capacitance C is set to various conditions within theabove-mentioned variation range. FIG. 7 shows seven kinds of examples asthe set conditions of the variation ΔPk of each parameter Pk.

Condition-1: All parameters Pk are set to the respective design valuesP0 k (center condition). The parasitic capacitance C calculated at thistime is Ctyp.

Condition-2: The variation of the film thickness d2 immediately abovethe interconnection layer M2 and the film thickness d1 immediately belowthe interconnection layer M2 each is set to the upper limit (+3σ).

Condition-3: The variation of the film thickness d2 immediately abovethe interconnection layer M2 and the film thickness d1 immediately belowthe interconnection layer M2 each is set to the lower limit (−3σ).

Condition-4: The variation of the film thickness d2 immediately abovethe interconnection layer M2 and the film thicknesses (d1, t1 and d0) ofrespective layers of the contact structure 20 each is set to the upperlimit (+3σ). That is, the variation of t1 and d0 each is set to theupper limit, in addition to the above-mentioned Condition-2.

Condition-5: The variation of the film thickness d2 immediately abovethe interconnection layer M2 and the film thicknesses (d1, t1 and d0) ofrespective layers of the contact structure 20 each is set to the lowerlimit (−3σ). That is, the variation of t1 and d0 each is set to thelower limit, in addition to the above-mentioned Condition-3.

Condition-δ6: The variation of the film thickness d2 immediately abovethe interconnection layer M2 and the film thickness d1 immediately belowthe interconnection layer M2 each is set to the lower limit (−3σ), as inthe case of the above Condition-3. In addition to that, the variation ofthe interconnect width w3 in the supra-interconnection layer M3 and theinterconnect width w1 in the sub-interconnection layer M1 each is set tothe upper limit (+3σ).

Condition-7: The variation of the film thickness d2 immediately abovethe interconnection layer M2 and the film thickness d1 immediately belowthe interconnection layer M2 each is set to the lower limit (−3σ), as inthe case of the above Condition-3. In addition to that, the variation ofthe interconnect thickness t3 in the supra-interconnection layer M3 andthe interconnect thickness t1 in the sub-interconnection layer M1 eachis set to the upper limit (+3σ).

FIG. 8 shows calculation results of the parasitic capacitance C withrespect to the above-mentioned conditions. The same trends as in thecase of FIG. 6 can be seen from FIG. 8.

2-3. Sensitivity Analysis System

The sensitivity analysis processing according to the present embodimentcan be achieved by a computer system. FIG. 9 is a block diagram showinga configuration of a sensitivity analysis system 100 executing thesensitivity analysis processing according to the present embodiment. Thesensitivity analysis system 100 has a memory device 110, a processor120, an input device 130 and a display device 140.

An interconnect structure data STR and a condition file CON are storedin the memory device 110. The interconnect structure data STR indicatesthe interconnect structure (layout, process cross-section) shown in FIG.4. The condition file CON indicates the various set conditions of thevariation ΔPk of each parameter Pk shown in FIG. 5 and FIG. 7.

A sensitivity analysis program PROG1 is a computer program executed bythe processor 120. The sensitivity analysis program PROG1 is stored inthe memory device 110. The sensitivity analysis program PROG1 may bestored in a computer-readable recording medium.

The processor 120 executes the sensitivity analysis program PROG1 toperform the sensitivity analysis according to the present embodiment.More specifically, a parameter setting unit 121, a capacitancecalculation unit 122, a sensitivity analysis unit 123 and a resultdisplay unit 124 are achieved by cooperation of the processor 120 andthe sensitivity analysis program PROG1. These function blocks performthe following processing.

FIG. 10 is a flow chart showing the sensitivity analysis processingaccording to the present embodiment.

Step S110:

First, the parameter setting unit 121 reads the interconnect structuredata STR and the condition file CON from the memory device 110.

Step S120:

Next, the parameter setting unit 121 sets the variation ΔPk of eachparameter Pk of the interconnect structure 1 to the respectiveconditions (refer to FIG. 5 and FIG. 7) described in the condition fileCON.

Step S130:

The capacitance calculation unit 122 includes the TCAD. The capacitancecalculation unit 122 performs the three-dimensional electromagneticanalysis of the interconnect structure 1 by using the parameter Pk setin Step S120 and thereby calculates the parasitic capacitance C of theinterconnect structure 1. The calculation of the parasitic capacitance Cis performed with respect to each of the above-mentioned plurality ofconditions. Consequently, the parasitic capacitances C of theinterconnect structure 1 in the respective conditions are calculated asshown in FIG. 6 or FIG. 8. The capacitance calculation unit 122 stores aresult data RST indicating the calculated parasitic capacitances C inthe memory device 110.

Step S140:

The sensitivity analysis unit 123 reads the result data RST from thememory device 110. Then, the sensitivity analysis unit 123 analyzesresponse of the parasitic capacitance C to the variation of theparameter Pk, based on the parasitic capacitances C calculated withrespect to the plurality of conditions. For example, the sensitivityanalysis unit 123 can analyze whether the parasitic capacitance Cincreases or decreases in response to the variation of each parameterPk, as mentioned above. That is, the sensitivity analysis unit 123 cananalyze variation trend of the parasitic capacitance C in response tothe variation of the parameter Pk. Moreover, the sensitivity analysisunit 123 calculates the variation of the parasitic capacitance C fromCtyp in each of the plurality of conditions. Then, the sensitivityanalysis unit 123 makes a comparison between the variation of theparasitic capacitance C and a predetermined threshold value (e.g.(C−Ctyp)/Ctyp=3%) to analyze the degree of contribution of eachparameter Pk to the parasitic capacitance C. It is thus possible toclassify the parameter Pk into the main parameter or the sub parameter.

Step S150:

The result display unit 124 reads the result data RST from the memorydevice 110 and has the display device 140 display the result of theanalysis. For example, the graph as shown in FIG. 6 and FIG. 8 isdisplayed by the display device 140.

The above-described results of the sensitivity analysis are utilized inCR extraction processing described below. As will be explained later, toutilize the results of the sensitivity analysis makes it possible toefficiently perform the CR extraction processing without deterioratingprecision.

3. CR Extraction in Consideration of Manufacturing. Variability

Next, extraction of the parasitic capacitance C and interconnectresistance R of the interconnect structure 1 will be described below.According to the present embodiment, the manufacturing variability istaken into consideration in the extraction processing. Meanwhile,unrealistic situation is excluded from the consideration. That is tosay, the “statistical relaxation” of the manufacturing variability isapplied.

3-1. Statistical Relaxation

First, concept of the statistical relaxation will be explained. As anexample, let us consider the manufacturing variability of parasiticcapacitance of a certain interconnection. Parameters contributing to theparasitic capacitance include an interconnect width w and aninterconnect thickness t of the interconnection, for example. A designvalue of the interconnect width w is w0, and variation from the designvalue w0 caused by the manufacturing variability is Δw. A design valueof the interconnect thickness t is t0, and variation from the designvalue t0 caused by the manufacturing variability is Δt.w=w0+Δwt=t0+Δt

Typically, a distribution of the variation Δw, Δt each is given by anormal distribution. When respective standard deviations are σw and σt,the variations Δw and Δt can be expressed as follows.Δw=αw×σwΔt=αt×σt

When the coefficient αw, αt each is within a range from −3 to +3,respective ranges of the variations Δw and Δt can be represented asfollows.Δw=−3σw to +3σwΔt=−3σt to +3σt

When the interconnect width w and the interconnect thickness t vary, theparasitic capacitance of the interconnection also varies accordingly. Inorder to search the maximum value and the minimum value of the parasiticcapacitance, the variations Δw and Δt may be separately varied withinthe above-mentioned respective ranges. However, since Δw and Δt areindependent variables that vary independently of each other, probabilitythat respective variation amplitudes of Δw and Δt simultaneously takethe maximum values is extremely low. For example, a case where Δw=−3σwand Δt=+3σt is unrealistic. Therefore, such extreme cases can beexcluded from consideration. That is, it is preferable that the maximumvalue and the minimum value of the parasitic capacitance are calculatedunder a condition that the respective variation amplitudes of Δw and Δtdo not simultaneously take the maximum values. This is the statisticalrelaxation.

An example of the statistical relaxation will be described withreference to FIG. 11. In FIG. 11, two axes orthogonal to each otherrepresent Δw and Δt, respectively. The origin O represents the centercondition (Δw=0, Δt=0), and a point (Δw, Δt) represents themanufacturing variation. A probability distribution of Δw, Δt each isgiven by a normal distribution. The range of Δw is from −3σw to +3σw,and the range of Δt is from −3σt to +3σt.

In the plane shown in FIG. 11, a “JPDF (Joint Probability DensityFunction)” that gives occurrence probability of the point (Δw, Δt) canbe defined. The JPDF may be called JDF (Joint Distribution Function). Acurve CEP shown in FIG. 11 represents a set of points (Δw, Δt) thatoccur with a same probability, which is hereinafter referred to as a“circle of equal probability”.

Since Δw and Δt are independent variables that vary independently ofeach other, probability that the respective variation amplitudes of Δwand Δt “simultaneously” take the maximum values is extremely low. Forexample, a point Q (Δw=+3σw, Δt=+3σt) shown in FIG. 11 is unrealistic.Therefore, such extreme cases are excluded from consideration, and onlyevents that occur with not less than a predetermined probability aretaken into consideration. It is preferable to use the above-mentionedcircle of equal probability CEP as the predetermined probability. Thatis to say, only points within the circle of equal probability CEP aretaken into consideration. Consequently, extreme cases such as the pointQ are excluded and the statistical relaxation is achieved. On searchingthe maximum value and the minimum value of the parasitic capacitance, Δwand Δt are set to points on the circle of equal probability CEP of theJPDF. In other words, the maximum value and the minimum value of theparasitic capacitance under a condition represented by the followingEquation (I) are calculated.[Equation. 1]√{square root over (αw ² +αt ²)}=3  (1)3-2. Parameter Setting Policy Regarding Interconnect Structure 1

In order to extract the parasitic capacitance C of the interconnectstructure 1 in consideration of the manufacturing variability, it isnecessary to vary each parameter Pk. However, to blindly vary all theparameter Pk is inefficient, which causes significant increase incomputation load. Therefore, according to the present embodiment, policyof setting the variation ΔPk of each parameter Pk is determined based onthe results of the sensitivity analysis described in the above Section2. The setting policy of the variation ΔPk of each parameter Pk in thepresent embodiment will be described with reference to FIG. 12.

As shown in FIG. 12, the interconnect structure 1 includes the maininterconnection 10 and the contact structure 20. The interconnect widthand the interconnect thickness of the main interconnection 10 arerepresented by the above-mentioned w and t, respectively. A layer placedimmediately above an interconnection layer in which the maininterconnection 10 is formed is the first layer. A layer placedimmediately below the interconnection layer in which the maininterconnection 10 is formed is the second layer. Third to n-th (n is aninteger equal to or more than 3) layers are placed in this order belowthe second layer. The contact structure 20 has a stacked structure inwhich conductors respectively formed in the second to the n-th layersare stacked. A film thickness of an interlayer insulating film formed inthe i-th layer (i=1 to n) is represented by “si”. A layer placedimmediately above the first layer is the supra-interconnection layer. Alayer placed immediately below the second layer is thesub-interconnection layer (third layer).

Parameters whose manufacturing variability is considered are as follows.As a result of the above-described sensitivity analysis, it turned outthat the variation of the interconnect widths of thesupra-interconnection layer and the sub-interconnection layer has littleeffect on the parasitic capacitance C. Therefore, the manufacturingvariability of the interconnect widths of the supra-interconnectionlayer and the sub-interconnection layer is not considered. Also, itturned out that the variation of the interconnect thickness of thesupra-interconnection layer has little effect on the parasiticcapacitance C. Therefore, the manufacturing variability of theinterconnect thickness of the supra-interconnection layer is notconsidered. Therefore, according to the present embodiment, themanufacturing variability of the interconnect width w and theinterconnect thickness t of the main interconnection 10, the filmthickness s1 of the first layer and the film thickness s2 to sn of therespective layers of the contact structure 20 is taken intoconsideration.

The variation of the film thickness si (i=1 to n) from its design valueis Δsi. The variation Δsi is represented within a predeterminedvariation range “−δsi to +δsi”. Here, −δsi is the lower limit of thevariation Δsi and +δsi is the upper limit of the variation Δsi.Typically, a distribution of the variation Δsi is given by a normaldistribution. When the standard deviation of the distribution is σsi,the variation Δsi can be represented by “Δsi=αsi×σsi”. When thecoefficient αsi is within a range from −3 to +3, the range of thevariation Δsi is represented by “−3σsi to +3σsi”, which is sufficientfrom a statistical point of view.

Parameters to which the statistical relaxation as shown in FIG. 11 isapplied are as follows. As an example, let us consider a case where thestatistical relaxation as shown in FIG. 11 is applied to all theparameters (w, t, si). In this case, the number of dimensions of thecircle of equal probability CEP needs to be increased. However, as thenumber of dimensions of the circle of equal probability CEP isincreased, the searching for the maximum value and the minimum value ofthe parasitic capacitance C becomes more complex, which causes increasein computation load. Therefore, according to the present embodiment, thenumber of dimensions of the circle of equal probability CEP is set assmall as possible without deteriorating precision. For that purpose, thestatistical relaxation is applied to the “main parameters” described inthe above Section 2. On the other hand, the statistical relaxation asshown in FIG. 11 is not applied to the “sub parameters”.

In the present embodiment, the main parameters include the interconnectwidth w and the interconnect thickness t of the main interconnection 10.Therefore, the statistical relaxation as shown in FIG. 11 is applied tothe variation Δw of the interconnect width w and the variation Δt of theinterconnect thickness t. That is, the maximum value and the minimumvalue of the interconnect capacitance C are searched for under acondition that respective variation amplitudes of the variations Δw andΔt do not simultaneously take maximum values (refer to the Equation(1)).

Meanwhile, the film thickness “si” of each layer is the sub parameter.Therefore, the statistical relaxation as shown in FIG. 11 is not appliedto the film thickness si. Instead, the variation Δsi of each filmthickness si is fixed to a predetermined value. That is, the maximumvalue and the minimum value of the interconnect capacitance C aresearched for under a condition that the variation Δsi is fixed to apredetermined value.

The setting value of the variation Δsi is represented by “+3σsi×γ” or“−3σsi×γ”. Here, γ is a relaxation coefficient, which is set to morethan than 0 and not more than 1. In a case of γ=1, it means the maximumamplitude within the predetermined variation range “−3σsi to +3σsi”.However, since the respective variations Δsi also are independentvariables that vary independently of each other, probability thatamplitudes of all the variations Δsi simultaneously take maximum valuesis extremely low. Therefore, the relaxation coefficient γ is preferablyset to less than 1. In this case, the manufacturing variability ismodestly relaxed. This means application of “simple statisticalrelaxation” instead of the statistical relaxation as shown in FIG. 11.An example of the relaxation coefficient γ less than 1 can be obtainedby the following equation (2) (refer to Japanese Laid-Open PatentApplication JP-2006-209702).

$\begin{matrix}\left\lbrack {{Equation}.\mspace{14mu} 2} \right\rbrack & \; \\{\gamma = \frac{\sqrt{\sum\limits_{i = 1}^{n}\;{\delta\;{si}^{2}}}}{\sum\limits_{i = 1}^{n}\;{\delta\;{si}}}} & (2)\end{matrix}$

It should be noted that the film thicknesses si of vias and contacts aredetermined by corresponding interlayer insulating films. The thicknessesof the interlayer insulating films are independent of each other. Aninterlayer insulating film corresponding to a via or a contact may beformed of a plurality of films. Even in that case, the statisticalrelaxation can be applied to respective film thicknesses of theplurality of films.

3-3. Extraction of Parasitic Capacitance C

Extraction of a maximum value Cmax and a minimum value Cmin of theparasitic capacitance C of the interconnect structure 1 is performed inconsideration of the manufacturing variability. FIG. 13 shows setconditions of the variation of each parameter from the design value atthe extraction processing.

(Center Condition)

In the center condition, the variations of the all parameters are set to0.

(Cmax Condition)

A Condition Under which the Maximum capacitance value Cmax is extractedis as follows. As to the main parameters, namely, the interconnect widthw and the interconnect thickness t of the main interconnection 10, thecondition that respective variation amplitudes of the variations Δw andΔt do not simultaneously take maximum values is imposed. Specifically,the condition represented by the above-mentioned Equation (1) isimposed. The variation Δs1 of the film thickness s1 immediately abovethe main interconnection 10 is set to a negative fixed value “−3σs1×γ”,and the variation Δs2 of the film thickness s2 immediately below themain interconnection 10 is set to a negative fixed value “−3σs2×γ”. Thisis because it has turned out from the result of the above-describedsensitivity analysis that the parasitic capacitance C is in inverseproportion to the film thickness s1, s2. Meanwhile, the variation Δsi ofthe film thickness si (i=3 to n) in each of the other layers belongingto the contact structure 20 is set to a positive fixed value “+3σsi×γ”.This is because it has turned out from the result of the above-describedsensitivity analysis that the parasitic capacitance C is proportional tothe film thickness si. A variation Δε of electric permittivity ε of theinterlayer insulating film 5 is set to a positive fixed value “+3σε”.Variations of the other parameters are set to 0.

(Cmin Condition)

A condition under which the minimum capacitance value Cmin is extractedis as follows. As to the main parameters, namely, the interconnect widthw and the interconnect thickness t of the main interconnection 10, thecondition that respective variation amplitudes of the variations Δw andΔt do not simultaneously take maximum values is imposed. Specifically,the condition represented by the above-mentioned Equation (1) isimposed. The variation Δs1 of the film thickness s1 immediately abovethe main interconnection 10 is set to a positive fixed value “+3σs1×γ”,and the variation Δs2 of the film thickness s2 immediately below themain interconnection 10 is set to a positive fixed value “+3σs2×γ”. Thisis because it has turned out from the result of the above-describedsensitivity analysis that the parasitic capacitance C is in inverseproportion to the film thickness s1, s2. Meanwhile, the variation Δsi ofthe film thickness si (i=3 to n) in each of the other layers belongingto the contact structure 20 is set to a negative fixed value “−3σsi×γ”.This is because it has turned out from the result of the above-describedsensitivity analysis that the parasitic capacitance C is proportional tothe film thickness si. The variation Δε of the electric permittivity εof the interlayer insulating film 5 is set to a negative fixed value“−3σε”. Variations of the other parameters are set to 0.

The parasitic capacitance C of the interconnect structure 1 iscalculated under each of the set conditions shown in FIG. 13. Here, theTOAD is used for calculating the parasitic capacitance C. That is tosay, the parasitic capacitance C of the interconnect structure 1 (wordinterconnect structure, bit interconnect structure) is calculatedthrough the “three-dimensional electromagnetic analysis” of the unitstructure shown in FIG. 4. By the three-dimensional electromagneticanalysis by using the TCAD, the parasitic capacitance C can becalculated with high precision. The parasitic capacitance C of theinterconnect structure 1 calculated under the center condition is Ctyp.

FIG. 14 shows the parasitic capacitance C calculated under theabove-mentioned Cmax condition and Cmin condition. The vertical axisrepresents the calculated parasitic capacitance C, and the horizontalaxis represents the angle θ (see FIG. 11) that specifies a point on thecircle of equal probability CEP. When the angle θ is changed, namely,when the variations Δw and Δt are changed under the conditionrepresented by the above Equation (1), the parasitic capacitance C alsois changed accordingly. In FIG. 14, a curve. V1 represents the change inthe parasitic capacitance C under the Cmax condition, and a curve V2represents the change in the parasitic capacitance C under the Cmincondition. The maximum point on the curve V1 corresponds to the maximumcapacitance value Cmax, and the minimum point on the curve V2corresponds to the minimum capacitance value Cmin. Note that apreferable method of searching for the maximum capacitance value Cmaxand the minimum capacitance value Cmin is described in JapaneseLaid-Open Patent Application JP-2009-283647 (not yet published on thepriority date) by the inventor of the present application.

In this manner, the center value Ctyp and the corner values Cmax andCmin are calculated (extracted) with regard to the parasitic capacitanceC of the interconnect structure 1. A correction parameter βcmax relatedto the maximum capacitance value Cmax and a correction parameter βcminrelated to the minimum capacitance value Cmin are respectivelycalculated as follows.βcmax=Cmax/Ctypβcmin=Cmin/Ctyp

In the case of the structure shown in FIG. 4, the above-described Ctyp,Cmax (βcmax) and Cmin (βcmin) are calculated with respect to each of theword interconnect structure including the word line WD and the bitinterconnect structure including the bit line BIT.

It should be noted that a coupling capacitance between the wordinterconnect structure and the bit interconnect structure is calculatedas follows. Coupling components of the maximum capacitance value Cmaxand the minimum capacitance value Cmin related to the word interconnectstructure are Ccp_wdmax and Ccp_wdmin, respectively. Similarly, couplingcomponents of the maximum capacitance value Cmax and the minimumcapacitance value Cmin related to the bit interconnect structure areCcp_bitmax and Ccp_bitmin, respectively. In this case, a maximum valueCcp_max of the coupling capacitance is the larger one of Ccp_wdmax andCcp_bitmax. A minimum value Ccp_min of the coupling capacitance is thesmaller one of Ccp_wdmin and Ccp_bitmin.

3-4. Extraction of Interconnect Resistance R

A method of calculating the interconnect resistance R of theinterconnect structure 1 according to the present embodiment will bedescribed below with reference to FIG. 15. The interconnect resistance Rof the interconnect structure 1 is expressed by a sum of a resistancevalue Rmain of the main interconnection 10 and a resistance valueRcontact of the contact structure 20.

$\begin{matrix}\left\lbrack {{Equation}.\mspace{14mu} 3} \right\rbrack & \; \\{R = {{{Rmain} + {Rcontact}} = {{Rmain} + {\sum\limits_{i = 2}^{n}\;{Ri}}}}} & (3)\end{matrix}$

The resistance value Rmain of the main interconnection 10 is calculatedin accordance with an equation: Rmain=ρ×1/(w×t). Here, ρ is electricalresistivity (unit: Ωm), and 1 is a length of the main interconnection10.

The resistance value Rcontact of the contact structure 20 is expressedby a sum of respective resistance values Ri (i=2 to n) of the layers ofthe contact structure 20. Here, the contact structure 20 is regarded asan interconnection extending in the Z-direction. Therefore, the filmthickness si (i=2 to n) of each layer of the contact structure 20corresponds to an interconnect length (length in the Z-direction) in theeach layer. When a cross-sectional area of the via is “Sv”, theresistance value Ri is calculated in accordance with an equation;Ri=ρ×si/Sv.

It should be noted that a resistance value of an interconnection(cushion interconnection) formed in an interconnection layer sandwichedbetween vias in the contact structure 20 is approximately calculatedbased on an envisioned current path. That is, as shown in FIG. 16, aresistance value R3 of an interconnection formed in an interconnectionlayer is calculated as a resistance value of an interconnection that hasthe cross-sectional area Sv and connects between the upper and lowervias. Here, positions of the upper and lower vias may be different fromeach other. Therefore, the resistance value R3 is calculated inaccordance with an equation: R3=ρ×s3/(Sv×cos φ), as shown in FIG. 16. Ina case where the upper and lower vias are aligned in the Z-direction(φ=0), the current path goes straight and thus the resistance value R3is calculated to be ρ×s3/Sv. The same applies to a case where thepositions of the upper and lower vias are different from each other inboth the X-direction and the Y-direction. When respective inclinationsfrom the Z-axis toward the X-direction and the Y-direction arerepresented by angles φx and φy, the resistance value R3 is calculatedin accordance with an equation: R3=ρ×s3/(Sv×cos ρx×cos φy).

Extraction of a maximum value Rmax and minimum value Rmin of theinterconnect resistance R of the interconnect structure 1 is performedin consideration of the manufacturing variability. Here, the parameterswhose manufacturing variability is considered include the interconnectwidth w and the interconnect thickness t of the main interconnection 10and the film thickness si (i=2 to n) of each layer of the contactstructure 20. FIG. 17 shows set conditions of the variation of eachparameter from the design value at the extraction processing.

(Center Condition)

In the center condition, the variations of the all parameters are set to0.

(Rmax Condition)

A condition under which the maximum resistance value Rmax is extractedis as follows. As to the main parameters, namely, the interconnect widthw and the interconnect thickness t of the main interconnection 10, thecondition that respective variation amplitudes of the variations Δw andΔt do not simultaneously take maximum values is imposed. Specifically,the condition represented by the above-mentioned Equation (1) isimposed. The variation Δsi of the film thickness si (i=2 to n) of eachlayer is set to a positive fixed value “+3σsi×γ”. Variations of theother parameters are set to 0.

(Rmin Condition)

a Condition Under which the Minimum resistance value Rmin is extractedis as follows. As to the main parameters, namely, the interconnect widthw and the interconnect thickness t of the main interconnection 10, thecondition that respective variation amplitudes of the variations Δw andΔt do not simultaneously take maximum values is imposed. Specifically,the condition represented by the above-mentioned Equation (1) isimposed. The variation Δsi of the film thickness si (i=2 to n) of eachlayer is set to a negative fixed value “−3σsi×γ”. Variations of theother parameters are set to 0.

The interconnect resistance R of the interconnect structure 1 iscalculated under each of the set conditions shown in FIG. 17. Theinterconnect resistance R of the interconnect structure 1 calculatedunder the center condition is Rtyp.

FIG. 18 shows the interconnect resistance R calculated under theabove-mentioned Rmax condition and Rmin condition. The vertical axisrepresents the calculated interconnect resistance R, and the horizontalaxis represents the angle θ (see FIG. 11) that specifies a point on thecircle of equal probability CEP. When the angle θ is changed, namely,when the variations Δw and Δt are changed under the conditionrepresented by the above Equation (1), the interconnect resistance Ralso is changed accordingly. In FIG. 18, a curve V3 represents thechange in the interconnect resistance R under the Rmax condition, and acurve V4 represents the change in, the interconnect resistance R underthe Rmin condition. The maximum point on the curve V3 corresponds to themaximum resistance value Rmax, and the minimum point on the curve V4corresponds to the minimum resistance value Rmin. Note that a preferablemethod of searching for the maximum resistance value Rmax and theminimum resistance value Rmin is described in Japanese Laid-Open PatentApplication JP-2009-283647 (not yet published on the priority date) bythe inventor of the present application.

In this manner, the center value Rtyp and the corner values Rmax andRmin are calculated (extracted) with regard to the interconnectresistance R of the interconnect structure 1. A correction parameterβrmax related to the maximum resistance value Rmax and a correctionparameter βrmin related to the minimum resistance value Rmin arerespectively calculated as follows.βrmax=Rmax/Rtypβrmin=Rmin/Rtyp

In the case of the structure shown in FIG. 4, the above-described Rtyp,Rmax (βrmax) and Rmin (βrmin) are calculated with respect to each of theword interconnect structure including the word line WD and the bitinterconnect structure including the bit line BIT,

3-5. Generation of CR-Added Cell Netlist

The interconnect structure 1 shown in FIG. 4 is for driving the celltransistor included in the memory cell. A netlist indicating aconnection relationship in the cell array is hereinafter referred to asa “cell netlist”. A “CR-added cell netlist” can be generated by addingthe parasitic capacitance C and the interconnect resistance R to a netcorresponding to the interconnect structure 1 in the cell netlist. Atthis time, the above-described center values Ctyp and Rtyp and thecorner values Cmax, Cmin, Rmax and Rmin are used. In general, theparasitic capacitance C and the interconnect resistance R have aninverse relationship. Therefore, when the maximum capacitance value Cmaxis added to the net, the minimum resistance value Rmin is added to thesame net. On the other hand, when the minimum capacitance value Cmin isadded to the net, the maximum resistance value Rmax is added to the samenet.

FIG. 19 shows various examples of the CR combination added to the wordinterconnect structure and the bit interconnect structure. In FIG. 19,Ctyp_wd, Cmax_wd, Cmin_wd, Rtyp wd, Rmax_wd and Rmin_wd are Ctyp, Cmax,Cmin, Rtyp, Rmax and Rmin of the word interconnect structure,respectively. Also, Ctyp_bit, Cmax_bit, Cmin_bit, Rtyp_bit, Rmax_bit andRmin_bit are Ctyp, Cmax, Cmin, Rtyp, Rmax and Rmin of the bitinterconnect structure, respectively.

By using the CR-added cell netlist, a circuit designer can performoperation verification (timing verification, delay verification) of thesemiconductor device. At this time, the operation verification of thesemiconductor device in a desired case can be performed by using theCR-added cell netlist corresponding to the desired case. For example,the various kinds of the CR-added cell netlists as shown in FIG. 19 areprovided as a library, and one corresponding to the desired case isappropriately selected from the library.

3-6. CR Extraction System

The CR extraction processing for the interconnect structure 1 accordingto the present embodiment can be achieved by a computer system. FIG. 20is a block diagram showing a configuration of a CR extraction system 200that executes the CR extraction processing according to the presentembodiment. The CR extraction system 200 has a memory device 210, aprocessor 220, an input device 230 and a display device 240.

An interconnect structure data STR, a capacitance condition file CCONand a resistance condition file RCON are stored in the memory device210. The interconnect structure data STR indicates the interconnectstructure (layout, process cross-section) shown in FIG. 4. Thecapacitance condition file CCON indicates the set conditions of thevariation of each parameter at the time of the capacitance extraction asshown in FIG. 13. The resistance condition file RCON indicates the setconditions of the variation of each parameter at the time of theresistance extraction as shown in FIG. 17.

A CR extraction program PROG2 is a computer program executed by theprocessor 220. The CR extraction program PROG2 is stored in the memorydevice 210. The CR extraction program PROG2 may be stored in acomputer-readable recording medium.

The processor 220 performs the CR extraction processing according to thepresent embodiment by executing the CR extraction program PROG2. Morespecifically, a capacitance extraction unit 221, a resistance extractionunit 222 and a cell netlist generation unit 223 are achieved bycooperation of the processor 220 and the CR extraction program PROG2.These function blocks perform the following processing.

FIG. 21 is a flow chart showing the CR extraction processing accordingto the present embodiment.

Step S210:

The capacitance extraction unit 221 reads the interconnect structuredata STR and the capacitance condition file CCON from the memory device210. Also, the resistance extraction unit 222 reads the interconnectstructure data STR and the resistance condition file RCON from thememory device 210.

Step S220:

The capacitance extraction unit 221 performs the extraction of theparasitic capacitance C of the interconnect structure 1 in accordancewith the statistical relaxation method of the present embodiment. Morespecifically, the capacitance extraction unit 221 sets the variation ofeach parameter of the interconnect structure 1 to each of the pluralityof conditions (refer to FIG. 13) described in the capacitance conditionfile CCON. The capacitance extraction unit 221 performs thethree-dimensional electromagnetic analysis of the interconnect structure1 by using the set parameters, and thereby calculates (extracts) thecenter value Ctyp and the corner values Cmax and Cmin of the parasiticcapacitance C of the interconnect structure 1.

Step S230:

The resistance extraction unit 222 performs the extraction of theinterconnect resistance R of the interconnect structure 1 in accordancewith the statistical relaxation method of the present embodiment. Morespecifically, resistance extraction unit 222 sets the variation of eachparameter of the interconnect structure 1 to each of the plurality ofconditions (refer to FIG. 17) described in the resistance condition fileRCON. The resistance extraction unit 222 uses the set parameters tocalculate (extract) the center value Rtyp and the corner values Rmax andRmin of the interconnect resistance R of the interconnect structure 1.

Step S240:

The cell netlist generation unit 223 generates the CR-added cell netlistCNET_CR by using the calculated center values Ctyp and Rtyp and cornervalues Cmax, Cmin, Rmax and Rmin. At this time, the cell netlistgeneration unit 223 may generate the CR-added cell netlists CNET_CR forvarious CR combinations as shown in FIG. 19. The cell netlist generationunit 223 stores the CR-added cell netlist CNET_CR in the memory device210.

4. Designing of Semiconductor Device

Next, a method of designing a semiconductor device will be described. Asan example, let us consider a semiconductor device including an SRAMmacro.

FIG. 22 shows a semiconductor device including an SRAM macro and aconfiguration of the SRAM macro. The SRAM macro is provided with a cellarray and a peripheral section. The cell array (CELL_ARRAY) has aplurality of memory cells CELL arranged in an array form as shown inFIG. 2 and functions as a data memory unit. The peripheral section isfor controlling an operation of the SRAM and is placed around the cellarray. The peripheral section includes a driver (DRIVER), a senseamplifier (SA), a data input/output unit (I/O), a controller (CNT) andso forth.

Circuit design processing according to the present embodiment can beachieved by a computer system. FIG. 23 is a block diagram showing aconfiguration of a circuit design system 300 executing the circuitdesign processing according to the present embodiment. The circuitdesign system 300 has a memory device 310, a processor 320, an inputdevice 330 and a display device 340.

A layout tool, an LPE tool, a netlist generation program, a delaycalculation tool, a timing analysis tool and the like are stored in thememory device 310. The netlist generation program includes a peripheralnetlist generation program PROG3 and a netlist synthesis program PROG4in addition to the above-mentioned CR extraction program PROG2. Thesetools and programs are computer programs executed by the processor 320.These tools and programs may be stored in a computer-readable recordingmedium. The processor 320 executes these tools and programs to performthe circuit design processing according to the present embodiment.

FIG. 24 is a flow chart showing the circuit design processing and amethod of manufacturing a semiconductor device according to the presentembodiment.

Step S310:

A netlist NET of the SRAM macro is generated through logic synthesis andso on. The netlist NET can be classified into the cell netlist that is anetlist of the cell array and a peripheral netlist that is a netlist ofthe peripheral section. The generated netlist NET is stored in thememory device 310. A netlist and a layout data of an SRAM cell may beseparately generated prior to the generation of the netlist NET of theSRAM macro.

Step S320:

The layout tool reads the netlist NET from the memory device 310 anddetermines a layout of the SRAM macro corresponding to the netlist NET.The layout can be classified into a layout of the cell array and alayout of the peripheral section. A layout data LAY indicating thedetermined layout is stored in the memory device 310. Moreover, theinterconnect structure data STR indicating the interconnect structureshown in FIG. 4 is generated based on the layout of the cell array andthe process cross-section. The generated interconnect structure data STRis stored in the memory device 310.

Step S330:

Next, the CR extraction processing for the SRAM macro is performed basedon the determined layout. As a result, the “CR-added netlist NET_CR” inwhich the parasitic capacitance C and the interconnect resistance R areadded to the netlist NET is generated.

According to a typical circuit design flow, the CR extraction processingis performed by the LPE tool. However, in the typical LPE tool,computation precision is deteriorated instead of improving processingspeed. On the other hand, according to the three-dimensionalelectromagnetic analysis by the use of the TCAD as described in thepresent embodiment, it is possible to extract the parasitic capacitanceC with high precision. Therefore, in the present embodiment, differentCR extraction methods are employed depending on the target. Morespecifically, as for the cell array, the high-precision CR extraction isperformed in accordance with the method described in the above Section3. As for the peripheral section, on the other hand, the high-speed CRextraction is performed by using the LPE tool.

Step S331;

The CR extraction program PROG2 extracts the parasitic capacitance C andthe interconnect resistance R of the interconnect structure 1 in thecell array in accordance with the method described in the above Section3. As a result, the CR-added cell netlist CNET_CR in which the extractedCR is added to the cell netlist is generated. The generated CR-addedcell netlist CNET_CR is stored in the memory device 310.

Step S332:

As for a random layout of the peripheral section, the parasiticcapacitance C and the interconnect resistance R are extracted throughthe LPE. More specifically, the peripheral netlist generation programPROG3 reads the layout data LAY from the memory device 310 and performsthe LPE based on the layout of the peripheral section. At this time, theperipheral netlist generation program PROG3 utilizes the LPE tool asappropriate. Moreover, it is preferable at this time that thestatistical relaxation is considered in accordance with the methoddescribed in Japanese Laid-Open Patent Application JP-2006-209702 orJapanese Laid-Open Patent Application JP-2008-028161. In this manner,the CR extraction for the peripheral section is performed through theLPE, and consequently a CR-added peripheral netlist PNET_CR in which theextracted CR is added to the peripheral netlist is generated. Thegenerated CR-added peripheral netlist PNET_CR is stored in the memorydevice 310.

Step S333:

The netlist synthesis program PROG4 reads the CR-added cell netlistCNET_CR and the CR-added peripheral netlist PNET_CR from the memorydevice 310. Then, the netlist synthesis program PROG4 combines theCR-added cell netlist CNET_CR and the CR-added peripheral netlistPNET_CR to generate the CR-added netlist NET_CR. The generated CR-addednetlist NET_CR is stored in the memory device 310.

Step S340:

Next, the operation verification such as the delay verification and thetiming verification is performed by using the generated CR-added netlistNET_CR. To this end, the delay calculation tool reads the CR-addednetlist NET_CR from the memory device 310 and calculates a delay valueof each net included in the CR-added netlist NET_CR. Then, the delaycalculation tool generates a delay file DELAY indicating the calculateddelay value of each net. The generated delay file DELAY is stored in thememory device 310.

Step S350:

The timing analysis tool reads the CR-added netlist NET_CR and the delayfile DELAY from the memory device 310. Then, the timing analysis toolperforms the timing analysis such as STA (Static Timing Analysis).

Step S360:

In a case where the timing analysis results in FAIL (Step S360; No), theprocessing returns back to the upstream Step S320 and modification ofthe layout is performed. Such returning-back is not preferable in termsof the design TAT. However, to consider the manufacturing variabilityduring the timing verification means that a condition to be met in thetiming verification becomes stricter. As the condition becomes stricter,the timing verification is more likely to result in FAIL and thus thenumber of circuit design modification times increases. This causesincrease in the design TAT.

In the CR extraction processing according to the present embodiment, themanufacturing variability is statistically relaxed and unrealisticextreme situations are excluded from the consideration. Thus, it is notnecessary to repeat circuit designing for supporting such extremesituations. It is therefore possible to perform the high-precisiontiming verification considering the manufacturing variability whilesuppressing increase in the design TAT.

Step S370:

In a case where the timing analysis results in PASS (Step S360; Yes), aconclusive layout data LAY is determined.

Step S380;

After that, the semiconductor device is manufactured based on thedetermined layout data LAY.

5. Summary

According to the present embodiment, influence of the manufacturingvariability of the contact structure 20 also is taken into considerationin the CR extraction processing. With increasing miniaturization of asemiconductor device in recent years, an interval between adjacentcontact structures 20 is getting narrower. Such tendency is conspicuous,for example, in a memory macro in which a large number of celltransistors are integrated. When the interval between adjacent contactstructures 20 becomes narrower, the influence of the manufacturingvariability of the contact structure 20 on the parasitic capacitance Cbecomes significant and nonnegligible. In these circumstances, thetechnique according to the present embodiment is particularly useful andenables the CR extraction with high precision in consideration of themanufacturing variability of the contact structure 20.

Moreover, in the CR extraction processing according to the presentembodiment, the manufacturing variability is statistically relaxed andunrealistic extreme situations are excluded from the consideration.Thus, it is not necessary to repeat circuit designing for supportingsuch extreme situations. It is possible to suppress increase in thedesign. TAT while considering the manufacturing variability.

Furthermore, according to the present embodiment, the sensitivityanalysis as described in the Section 2 is performed beforehand. In theCR extraction processing, the variation ΔPk of each parameter Pk issuitably set based on the results of the sensitivity analysis. Forexample, the statistical relaxation as shown in FIG. 11 is applied onlyto the main parameters that greatly contribute to the parasiticcapacitance C. On the other hand, the variation of the sub parameterless contributing to the parasitic capacitance C is set to apredetermined fixed value. Here, plus and minus of the predeterminedfixed value also is suitably set based on the results of the sensitivityanalysis. In this manner, unnecessary increase in computation load inthe CR extraction processing can be prevented without deteriorating theprecision. That is, both of the precision and the design TAT can besatisfied.

Furthermore, according to the present embodiment, different CRextraction methods are employed for different targets. Specifically, asfor the cell array, the high-precision CR extraction is performed byusing the TCAD (refer to the Section 3). As for a random layout in theperipheral section, on the other hand, the high-speed CR extraction isperformed through the LPE. As a result, both of the precision and thedesign TAT can be satisfied.

In the above description, the SRAM is used as an example. The sameapplies to DRAM, eDRAM, flash memory and the like.

The embodiments of the present invention have been described above withreference to the attached drawings. It is apparent that the presentinvention is not limited to the above embodiments and may be modifiedand changed without departing from the scope and spirit of theinvention.

1. A sensitivity analysis system comprising: a memory device in which aninterconnect structure data indicating an interconnect structureincluded in a semiconductor device is stored, wherein said interconnectstructure comprises: a main interconnection formed in an interconnectionlayer; and a contact structure electrically connected to said maininterconnection and extending from said main interconnection toward asemiconductor substrate, wherein a plurality of parameters contribute toparasitic capacitance of said interconnect structure, and variation ofeach of said plurality of parameters from a design value caused bymanufacturing variability is represented within a predetermined range; aparameter setting unit configured to set said variation to a pluralityof conditions within said predetermined range; a capacitance calculationunit configured to calculate said parasitic capacitance of saidinterconnect structure in each of said plurality of conditions; and asensitivity analysis unit configured to analyze, based on saidcalculated parasitic capacitance, response of said parasitic capacitanceto variation of said each parameter, wherein said sensitivity analysisunit calculates variation of said parasitic capacitance from a designvalue in each of said plurality of conditions, and said sensitivityanalysis unit makes a comparison between the variation of said parasiticcapacitance and a predetermined threshold value to analyze degree ofcontribution of said each parameter to said parasitic capacitance. 2.The sensitivity analysis system according to claim 1, wherein saidsensitivity analysis unit analyzes whether said parasitic capacitanceincreases or decreases in response to the variation of said eachparameter.
 3. The sensitivity analysis system according to claim 1,wherein said semiconductor device comprises a memory, said interconnectstructure drives a cell transistor included in a memory cell of saidmemory, and said main interconnection is a word line or a bit line thatis connected to said cell transistor through said contact structure. 4.A sensitivity analysis system comprising: a memory device in which aninterconnect structure data indicating an interconnect structureincluded in a semiconductor device is stored, wherein said interconnectstructure comprises: a main interconnection formed in an interconnectionlayer; and a contact structure electrically connected to said maininterconnection and extending from said main interconnection toward asemiconductor substrate, wherein a plurality of parameters contribute toparasitic capacitance of said interconnect structure, and variation ofeach of said plurality of parameters from a design value caused bymanufacturing variability is represented within a predetermined range; aparameter setting unit configured to set said variation to a pluralityof conditions within said predetermined range; a capacitance calculationunit configured to calculate said parasitic capacitance of saidinterconnect structure in each of said plurality of conditions; and asensitivity analysis unit configured to analyze, based on saidcalculated parasitic capacitance, response of said parasitic capacitanceto variation of said each parameter, wherein a layer placed immediatelyabove said interconnection layer is a first layer, and a layer placedimmediately below said interconnection layer is a second layer, whereinsaid plurality of parameters comprise: a first film thickness that is afilm thickness of an interlayer insulating film in said first layer; anda second film thickness that is a film thickness of an interlayerinsulating film in said second layer, wherein the variation of saidfirst film thickness from the design value is represented within a rangefrom −δ1 to +δ1, and the variation of said second film thickness fromthe design value is represented within a range from −δ2 to +δ2, whereinsaid plurality of conditions comprise: a first condition in which saidplurality of parameters all are set to the design values; a secondcondition in which the variation of said first film thickness and saidsecond film thickness are set to +δ1 and +δ2, respectively; and a thirdcondition in which the variation of said first film thickness and saidsecond film thickness are set to −δ1 and −δ2, respectively.
 5. Thesensitivity analysis system according to claim 4, wherein said contactstructure has a stacked structure in which conductors formed inrespective layers below said interconnection layer are stacked, whereinsaid plurality of parameters further comprise a third film thicknessthat is a film thickness of an interlayer insulating film in each layeramong said respective layers other than said second layer, and thevariation of said third film thickness from the design value isrepresented within a range from −δ3 to +δ3, and wherein said pluralityof conditions further comprise: a fourth condition in which thevariation of said first film thickness, said second film thickness andsaid third film thickness are set to +δ1, +δ2 and +δ3, respectively; anda fifth condition in which the variation of said first film thickness,said second film thickness and said third film thickness are set to −δ1,−δ2 and −δ3, respectively.
 6. The sensitivity analysis system accordingto claim 5, wherein a layer placed immediately above said first layer isa supra-interconnection layer, and a layer placed immediately below saidsecond layer is a sub-interconnection layer, wherein said plurality ofparameters further comprise: a first interconnect width and a firstinterconnect thickness that respectively are a width and a thickness ofan interconnection formed in said supra-interconnection layer; and asecond interconnect width and a second interconnect thickness thatrespectively are a width and a thickness of an interconnection formed insaid sub-interconnection layer, wherein the variation of said firstinterconnect width from the design value is represented within a rangefrom −δ4 to +δ4, the variation of said first interconnect thickness fromthe design value is represented within a range from −δ5 to +δ5, thevariation of said second interconnect width from the design value isrepresented within a range from −δ6 to +δ6, and the variation of saidsecond interconnect thickness from the design value is representedwithin a range from −δ7 to +δ7, and wherein said plurality of conditionsfurther comprise: a sixth condition in which the variation of said firstfilm thickness, said second film thickness, said first interconnectwidth and said second interconnect width are set to −δ1, −δ2, +δ4 and+δ6, respectively; and a seventh condition in which the variation ofsaid first film thickness, said second film thickness, said firstinterconnect thickness and said second interconnect thickness are set to−δ1, −δ2, +δ5 and +δ7, respectively.
 7. A non-transient computerreadable medium storing a sensitivity analysis program that, whenexecuted by a computer, causes the computer to perform a methodcomprising: reading an interconnect structure data indicating aninterconnect structure included in a semiconductor device from a memorydevice, wherein said interconnect structure comprises: a maininterconnection formed in an interconnection layer; and a contactstructure electrically connected to said main interconnection andextending from said main interconnection toward a semiconductorsubstrate, wherein a plurality of parameters contribute to parasiticcapacitance of said interconnect structure, and variation of each ofsaid plurality of parameters from a design value caused by manufacturingvariability is represented within a predetermined range; setting saidvariation to a plurality of conditions within said predetermined range;calculating said parasitic capacitance of said interconnect structure ineach of said plurality of conditions; analyzing, based on saidcalculated parasitic capacitance, response of said parasitic capacitanceto variation of said each parameter; calculating a variation of saidparasitic capacitance from a design value in each of said plurality ofconditions; and comparing between the variation of said parasiticcapacitance and a predetermined threshold value to analyze degree ofcontribution of said each parameter to said parasitic capacitance.